scripts/optimization/lsqnonneg.m

## Copyright (C) 2008 Bill Denney
## Copyright (C) 2008 Jaroslav Hajek
## Copyright (C) 2009 VZLU Prague
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {@var{x} =} lsqnonneg (@var{c}, @var{d})
## @deftypefnx {Function File} {@var{x} =} lsqnonneg (@var{c}, @var{d}, @var{x0})
## @deftypefnx {Function File} {[@var{x}, @var{resnorm}] =} lsqnonneg (@dots{})
## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}] =} lsqnonneg (@dots{})
## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}] =} lsqnonneg (@dots{})
## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}] =} lsqnonneg (@dots{})
## @deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}] =} lsqnonneg (@dots{})
## Minimize @code{norm (@var{c}*@var{x}-d)} subject to @code{@var{x} >=
## 0}.  @var{c} and @var{d} must be real.  @var{x0} is an optional
## initial guess for @var{x}.
##
## Outputs:
## @itemize @bullet
## @item resnorm
##
## The squared 2-norm of the residual: norm(@var{c}*@var{x}-@var{d})^2
## @item residual
##
## The residual: @var{d}-@var{c}*@var{x}
## @item exitflag
##
## An indicator of convergence.  0 indicates that the iteration count
## was exceeded, and therefore convergence was not reached; >0 indicates
## that the algorithm converged.  (The algorithm is stable and will
## converge given enough iterations.)
## @item output
##
## A structure with two fields:
## @itemize @bullet
## @item "algorithm": The algorithm used ("nnls")
## @item "iterations": The number of iterations taken.
## @end itemize
## @item lambda
##
## Not implemented.
## @end itemize
## @seealso{optimset, pqpnonneg}
## @end deftypefn

## PKG_ADD: __all_opts__ ("lsqnonneg");

## This is implemented from Lawson and Hanson's 1973 algorithm on page
## 161 of Solving Least Squares Problems.

function [x, resnorm, residual, exitflag, output, lambda] = lsqnonneg (c, d, x = [], options = struct ())

  if (nargin == 1 && ischar (c) && strcmp (c, 'defaults'))
    x = optimset ("MaxIter", 1e5);
    return
  endif

  if (! (nargin >= 2 && nargin <= 4 && ismatrix (c) && ismatrix (d) && isstruct (options)))
    print_usage ();
  endif

  ## Lawson-Hanson Step 1 (LH1): initialize the variables.
  m = rows (c);
  n = columns (c);
  if (isempty (x))
    ## Initial guess is 0s.
    x = zeros (n, 1);
  else
    ## ensure nonnegative guess.
    x = max (x, 0);
  endif

  useqr = m >= n;
  max_iter = optimget (options, "MaxIter", 1e5);

  ## Initialize P, according to zero pattern of x.
  p = find (x > 0).';
  if (useqr)
    ## Initialize the QR factorization, economized form.
    [q, r] = qr (c(:,p), 0);
  endif

  iter = 0;

  ## LH3: test for completion.
  while (iter < max_iter)
    while (iter < max_iter)
      iter++;

      ## LH6: compute the positive matrix and find the min norm solution
      ## of the positive problem.
      if (useqr)
        xtmp = r \ q'*d;
      else
        xtmp = c(:,p) \ d;
      endif
      idx = find (xtmp < 0);

      if (isempty (idx)) 
        ## LH7: tmp solution found, iterate.
        x(:) = 0;
        x(p) = xtmp;
        break;
      else
        ## LH8, LH9: find the scaling factor.
        pidx = p(idx);
        sf = x(pidx)./(x(pidx) - xtmp(idx));
        alpha = min (sf);
        ## LH10: adjust X.
        xx = zeros (n, 1);
        xx(p) = xtmp;
        x += alpha*(xx - x);
        ## LH11: move from P to Z all X == 0.
        ## This corresponds to those indices where minimum of sf is attained.
        idx = idx (sf == alpha);
        p(idx) = [];
        if (useqr)
          ## update the QR factorization.
          [q, r] = qrdelete (q, r, idx);
        endif
      endif
    endwhile
      
    ## compute the gradient.
    w = c'*(d - c*x);
    w(p) = [];
    if (! any (w > 0))
      if (useqr)
        ## verify the solution achieved using qr updating.
        ## in the best case, this should only take a single step.
        useqr = false;
        continue;
      else
        ## we're finished.
        break;
      endif
    endif

    ## find the maximum gradient.
    idx = find (w == max (w));
    if (numel (idx) > 1)
      warning ("lsqnonneg:nonunique",
               "A non-unique solution may be returned due to equal gradients.");
      idx = idx(1);
    endif
    ## move the index from Z to P. Keep P sorted.
    z = [1:n]; z(p) = [];
    zidx = z(idx);
    jdx = 1 + lookup (p, zidx);
    p = [p(1:jdx-1), zidx, p(jdx:end)];
    if (useqr)
      ## insert the column into the QR factorization.
      [q, r] = qrinsert (q, r, jdx, c(:,zidx));
    endif

  endwhile
  ## LH12: complete.

  ## Generate the additional output arguments.
  if (nargout > 1)
    resnorm = norm (c*x - d) ^ 2;
  endif
  if (nargout > 2)
    residual = d - c*x;
  endif
  exitflag = iter;
  if (nargout > 3 && iter >= max_iter)
    exitflag = 0;
  endif
  if (nargout > 4)
    output = struct ("algorithm", "nnls", "iterations", iter);
  endif
  if (nargout > 5)
    lambda = zeros (size (x));
    lambda(p) = w;
  endif

endfunction

## Tests
%!test
%! C = [1 0;0 1;2 1];
%! d = [1;3;-2];
%! assert (lsqnonneg (C, d), [0;0.5], 100*eps)

%!test
%! C = [0.0372 0.2869;0.6861 0.7071;0.6233 0.6245;0.6344 0.6170];
%! d = [0.8587;0.1781;0.0747;0.8405];
%! xnew = [0;0.6929];
%! assert (lsqnonneg (C, d), xnew, 0.0001)